ok...you have some $1 bills, $5 bills, and $10 bills. There are 225 bills total and they are worth $596. If you have seven times as many $1 as you do $10, how many of each bill do you have?????

i have tried this problem over 20 times!!!!!!

Sincerly
armath

You have 2 equations here, let "A" stand for the number of 1 dollar bills you have, "B" stand for the number of 5 dollar bills you have, and C stand for the number of 10 dollar bills you have.

Then you know that the total number of bills is 225, so A + B + C = 225

And you know that each A is worth 1 dollar, each B is worth 5 dollars, and each C is worth 10 dollars, and that you have 596 dollars total. So A + 5B + 10C = 596

So set the equations next to eachother:

Now, we have three variables, but only 2 equations. But we know that there are 7 times as many ones as tens. This means that the number of ones is equal to 7 times the number of tens. (Think if there was 1 ten, then there would need to be 7 ones) so A = 7C. This means we can substitute out A:

Simplify

Now you can solve this any way you like, I'll do it with substitution. Since B + 8C = 225, it we can see that B = 225-8C So lets substitute that into our other equation:

Now we can plug that back into our other equations:

And simplify

Again, you can solve this any way you like, I'll use matrices this time:

Subtract line 1 from line 2:

Divide line 2 by 4

And substitute The value of B into the first equation:

Subtract 41 from line 1

So we have A=161, B=41, and C=23

We can verify our answer by plugging them back into our equations:

You can use your calculator to verify that these answers are correct (they are, I checked them, but you should get into that practice).

And so we only need to verify that they also meet the last criteria that A=7C, so we plug the values in: 161=7(23) and we see that this is correct as well. So we know that our answer is correct.

Some parting advice: Make sure you set them up right, ie seven times as many ones as tens might make you want to say 7A=C, but plug in a value for A, and you will see that C is actually 7 times bigger than A, so it must be the other way around that A=7C. Second, when manipulating equations, use one to manipulate the other, you can never use an equation to manipulate itself, because you will end up with A=A or 0=0 or some other such useless equation. So if you solve for a variable in one equation, use that on the other equation.